Compactly assembled category

Introduction

Compactly assembled categories are the right generality for modern applications where compact generation fails but the category still feels “finite-dimensional”. The motivating example: $\mathsf{Shv}(X)$ for $X$ locally compact Hausdorff is not compactly generated, yet behaves like a compactly generated category for nearly every K-theoretic and trace-theoretic purpose.

The intuition: an object of a compactly assembled category is built out of compactly exhaustible atoms — sequences whose transition maps are “compact” in a precise sense. Compactness is now a property of morphisms, not just objects. In the stable world, this generality is forced on us: compactly assembled is precisely dualizable in $\mathsf{Pr}^L_{\mathrm{st}}$, which is what we need for K-theory, traces, and the symmetric monoidal structure to behave.

Body

A presentable category $\mathcal{C}$ is compactly assembled if it is generated under colimits by compactly exhaustible objects — objects of the form $\operatorname*{colim}_n X_n$ along a sequence $X_0 \to X_1 \to \cdots$ in which every transition map is a compact morphism.

A compact morphism $f\colon X \to Y$ is one for which, for every filtered colimit $Z \simeq \operatorname*{colim}_i Z_i$, the natural square

\[ \begin{array}{ccc} \operatorname*{colim}_i \operatorname{Hom}(Y, Z_i) & \to & \operatorname*{colim}_i \operatorname{Hom}(X, Z_i) \\ \downarrow & & \downarrow \\ \operatorname{Hom}(Y, Z) & \to & \operatorname{Hom}(X, Z) \end{array} \]

is a pullback.

Lurie–Clausen characterisation

For a presentable $\mathcal{C}$, the following are equivalent:

$\mathcal{C}$ is compactly assembled.
There is a regular cardinal $\kappa$ such that $\mathcal{C}$ is $\kappa$-presentable and the colimit functor $k\colon \mathsf{Ind}(\mathcal{C}^{\kappa}) \to \mathcal{C}$ has a left adjoint $\hat y$.
$\mathcal{C}$ is a retract in $\mathsf{Pr}^L$ of a Compactly generated category.
Filtered colimits commute with all small limits in $\mathcal{C}$.

In the stable world, $\mathcal{C} \in \mathsf{Pr}^L_{\mathrm{st}}$ is compactly assembled iff it is a dualizable object of $\mathsf{Pr}^L_{\mathrm{st}}$.

Why care

$\mathsf{Shv}(X)$ for $X$ locally compact Hausdorff is compactly assembled but not in general compactly generated.
Efimov K-theory extends K-theory from compactly generated to compactly assembled categories, recovering “classical” K-theory of categories like $\mathsf{Shv}(X, \mathsf{Sp})$.
In $\mathsf{Pr}^L_{\mathrm{ca}}$, dualizability gives an analogue of finite-dimensional vector spaces, with internal Hom and a meaningful trace.

Relation to other notions

Every Compactly generated category is compactly assembled.
Every compactly assembled category is a Presentable category, $\aleph_1$-presentable in fact.
A Stable category is compactly assembled iff it is dualizable in $\mathsf{Pr}^L_{\mathrm{st}}$.

Pointers

Krause–Nikolaus–Pützstück, Sheaves on Manifolds, §2.3
Ramzi, Dualizable presentable ∞-categories, arXiv:2410.21537
Efimov, K-theory and localizing invariants of large categories, arXiv:2405.12169